IsotropicState - Revision history

Nathaniel at 12:50, 23 September 2014

2014-09-23T12:50:40Z

← Older revision		Revision as of 12:50, 23 September 2014
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	<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. ''Phys. Rev. A'', 59:4206–4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>		<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. ''Phys. Rev. A'', 59:4206–4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>
	<center><math>\rho_\alpha := (1-\alpha~~)I/~~d^2 + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>		<center><math>\displaystyle\rho_\alpha := \frac{1-\alpha}{d^2}I\otimes I + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>
	where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.		where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.

Nathaniel at 12:47, 23 September 2014

2014-09-23T12:47:00Z

← Older revision		Revision as of 12:47, 23 September 2014
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	<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. ''Phys. Rev. A'', 59:4206--4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>		<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. ''Phys. Rev. A'', 59:4206–4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>
	<center><math>\rho_\alpha := (1-\alpha)I/d^2 + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>		<center><math>\rho_\alpha := (1-\alpha)I/d^2 + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>
	where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.		where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.

Nathaniel at 12:46, 23 September 2014

2014-09-23T12:46:35Z

← Older revision		Revision as of 12:46, 23 September 2014
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	\|v=0.50}}		\|v=0.50}}
	<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A, 59:4206--4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>		<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. ''Phys. Rev. A'', 59:4206--4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>
	<center><math>\rho_\alpha := (1-\alpha)I/d^2 + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>		<center><math>\rho_\alpha := (1-\alpha)I/d^2 + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>
	where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.		where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.

Nathaniel at 12:45, 23 September 2014

2014-09-23T12:45:17Z

← Older revision		Revision as of 12:45, 23 September 2014
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	\|desc=Produces an [[isotropic state]]		\|desc=Produces an [[isotropic state]]
	\|rel=[[WernerState]]		\|rel=[[WernerState]]
			\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]
	\|upd=September 22, 2014		\|upd=September 22, 2014
	\|v=0.50}}		\|v=0.50}}

Nathaniel: /* A qutrit isotropic state */

2014-09-22T14:33:01Z

A qutrit isotropic state

← Older revision		Revision as of 14:33, 22 September 2014
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	Isotropic states in general have a lot of zero entries, so this function always returns a sparse matrix. If you want a full matrix (as above), use MATLAB's [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/full.html full] function.		Isotropic states in general have a lot of zero entries, so this function always returns a sparse matrix. If you want a full matrix (as above), use MATLAB's [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/full.html full] function.
	<syntaxhighlight>		<syntaxhighlight>
	>> IsotropicState(3,1/2,1)		>> IsotropicState(3,1/2)

	ans =		ans =

Nathaniel at 14:29, 22 September 2014

2014-09-22T14:29:24Z

← Older revision		Revision as of 14:29, 22 September 2014
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	\|name=IsotropicState		\|name=IsotropicState
	\|desc=Produces an [[isotropic state]]		\|desc=Produces an [[isotropic state]]
	~~\|req=[[iden]]<br />[[MaxEntangled]]<br />[[opt_args]]~~
	\|rel=[[WernerState]]		\|rel=[[WernerState]]
	\|upd=~~January 7~~, ~~2013~~		\|upd=September 22, 2014
	\|v=1.00}}		\|v=0.50}}
	<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns ~~the~~ [[isotropic state]] ~~requested by~~ the ~~user~~. ~~The isotropic~~ state ~~may be either full or~~ sparse.		<tt>'''IsotropicState'''</tt> is a [[List of functions\|function]] that returns an [[isotropic state]] (i.e., a state of the following form):<ref name="HH97">M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A, 59:4206--4216, 1999. E-print: [http://arxiv.org/abs/quant-ph/9708015 arXiv:quant-ph/9708015]</ref>
			<center><math>\rho_\alpha := (1-\alpha)I/d^2 + \alpha\|\psi_+\rangle\langle\psi_+\| \in M_d \otimes M_d,</math></center>
			where $\|\psi_+\rangle:=\frac{1}{\sqrt{d}}\sum_j\|j\rangle\otimes\|j\rangle$ is the standard [[MaxEntangled\|maximally-entangled pure state]]. Note that the output of this function is a sparse matrix.

	==Syntax==		==Syntax==
	* <tt>RHO = IsotropicState(DIM,ALPHA)</tt>		* <tt>RHO = IsotropicState(DIM,ALPHA)</tt>
	* <tt>RHO = IsotropicState(DIM,ALPHA,SP)</tt>

	==Argument descriptions==		==Argument descriptions==
	* <tt>DIM</tt>: Dimension of the local subsystems on which <tt>RHO</tt> acts.		* <tt>DIM</tt>: Dimension of the local subsystems on which <tt>RHO</tt> acts.
	* <tt>ALPHA</tt>: A parameter that specifies which [[isotropic state]] is to be returned. In particular, <tt>RHO = (1-ALPHA)I/DIM^2 + ALPHAE</tt>, where <tt>I</tt> is the identity operator and <tt>E</tt> is the projection onto the standard maximally-entangled pure state on two copies of <tt>DIM</tt>-dimensional space. In order for <tt>RHO</tt> to be positive semidefinite (and hence a valid density matrix), it must be the case that <tt>-1/(DIM^2-1) ≤ ALPHA ≤ 1</tt>.		* <tt>ALPHA</tt>: A parameter that specifies which isotropic state is to be returned. In particular, <tt>RHO = (1-ALPHA)I/DIM^2 + ALPHAE</tt>, where <tt>I</tt> is the identity operator and <tt>E</tt> is the projection onto the standard maximally-entangled pure state on two copies of <tt>DIM</tt>-dimensional space. In order for <tt>RHO</tt> to be positive semidefinite (and hence a valid density matrix), it must be the case that <tt>-1/(DIM^2-1) ≤ ALPHA ≤ 1</tt>.
	* <tt>SP</tt> (optional, default 0): A flag (either 1 or 0) indicating that the isotropic state produced should or should not be sparse.

	==Examples==		==Examples==
	===A qutrit isotropic state===		===A qutrit isotropic state===
	To generate the isotropic state with parameter $\alpha = 1/2$, the following code suffices:		To generate the isotropic state with parameter $\alpha = 1/2$, the following code suffices:
	<~~pre~~>		<syntaxhighlight>
	>> IsotropicState(3,1/2)		>> full(IsotropicState(3,1/2))

	ans =		ans =
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	0 0 0 0 0 0 0 0.0556 0		0 0 0 0 0 0 0 0.0556 0
	0.1667 0 0 0 0.1667 0 0 0 0.2222		0.1667 0 0 0 0.1667 0 0 0 0.2222
	</~~pre~~>		</syntaxhighlight>

	Isotropic states in general have a lot of zero entries, so you ~~will usually save~~ a ~~lot of memory by specifying <tt>SP = 1</tt>~~, ~~which causes the isotropic state that is generated to be sparse~~:		Isotropic states in general have a lot of zero entries, so this function always returns a sparse matrix. If you want a full matrix (as above), use MATLAB's [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/full.html full] function.
	<~~pre~~>		<syntaxhighlight>
	>> IsotropicState(3,1/2,1)		>> IsotropicState(3,1/2,1)

Line 57:		Line 56:
	(5,9) 0.1667		(5,9) 0.1667
	(9,9) 0.2222		(9,9) 0.2222
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=IsotropicState}}

			==References==
			<references />

Nathaniel: Created page with "{{Function |name=IsotropicState |desc=Produces an isotropic state |req=iden
MaxEntangled
opt_args |rel=WernerState |upd=January 7, 2013 |v=1.00}}..."

2013-01-08T19:34:15Z

Created page with "{{Function |name=IsotropicState |desc=Produces an isotropic state |req=iden<br />MaxEntangled<br />opt_args |rel=WernerState |upd=January 7, 2013 |v=1.00}}..."

New page

{{Function
|name=IsotropicState
|desc=Produces an [[isotropic state]]
|req=[[iden]]<br />[[MaxEntangled]]<br />[[opt_args]]
|rel=[[WernerState]]
|upd=January 7, 2013
|v=1.00}}
<tt>'''IsotropicState'''</tt> is a [[List of functions|function]] that returns the [[isotropic state]] requested by the user. The isotropic state may be either full or sparse.

==Syntax==
* <tt>RHO = IsotropicState(DIM,ALPHA)</tt>
* <tt>RHO = IsotropicState(DIM,ALPHA,SP)</tt>

==Argument descriptions==
* <tt>DIM</tt>: Dimension of the local subsystems on which <tt>RHO</tt> acts.
* <tt>ALPHA</tt>: A parameter that specifies which [[isotropic state]] is to be returned. In particular, <tt>RHO = (1-ALPHA)*I/DIM^2 + ALPHA*E</tt>, where <tt>I</tt> is the identity operator and <tt>E</tt> is the projection onto the standard maximally-entangled pure state on two copies of <tt>DIM</tt>-dimensional space. In order for <tt>RHO</tt> to be positive semidefinite (and hence a valid density matrix), it must be the case that <tt>-1/(DIM^2-1) ≤ ALPHA ≤ 1</tt>.
* <tt>SP</tt> (optional, default 0): A flag (either 1 or 0) indicating that the isotropic state produced should or should not be sparse.

==Examples==
===A qutrit isotropic state===
To generate the isotropic state with parameter $\alpha = 1/2$, the following code suffices:
<pre>
>> IsotropicState(3,1/2)

ans =

0.2222 0 0 0 0.1667 0 0 0 0.1667
0 0.0556 0 0 0 0 0 0 0
0 0 0.0556 0 0 0 0 0 0
0 0 0 0.0556 0 0 0 0 0
0.1667 0 0 0 0.2222 0 0 0 0.1667
0 0 0 0 0 0.0556 0 0 0
0 0 0 0 0 0 0.0556 0 0
0 0 0 0 0 0 0 0.0556 0
0.1667 0 0 0 0.1667 0 0 0 0.2222
</pre>

Isotropic states in general have a lot of zero entries, so you will usually save a lot of memory by specifying <tt>SP = 1</tt>, which causes the isotropic state that is generated to be sparse:
<pre>
>> IsotropicState(3,1/2,1)

ans =

(1,1) 0.2222
(5,1) 0.1667
(9,1) 0.1667
(2,2) 0.0556
(3,3) 0.0556
(4,4) 0.0556
(1,5) 0.1667
(5,5) 0.2222
(9,5) 0.1667
(6,6) 0.0556
(7,7) 0.0556
(8,8) 0.0556
(1,9) 0.1667
(5,9) 0.1667
(9,9) 0.2222
</pre>